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Let G {\displaystyle G} be a unimodular locally profinite group such that G / K {\displaystyle G/K} is at most countable for all open compact subgroups K, and μ {\displaystyle \mu } a left Haar measure on G {\displaystyle G} . Let C c ∞ ( G ) {\displaystyle C_{c}^{\infty }(G)} denote the space of locally constant funct... | Wikipedia - Locally profinite group | null | null | null |
The algebra plays an important role in the study of smooth representations of locally profinite groups. Indeed, one has the following: given a smooth representation ( ρ , V ) {\displaystyle (\rho ,V)} of G, we define a new action on V: ρ ( f ) = ∫ G f ( g ) ρ ( g ) d μ ( g ) . {\displaystyle \rho (f)=\int _{G}f(g)\rho ... | Wikipedia - Locally profinite group | null | null | null |
Thus, we have the functor ρ ↦ ρ {\displaystyle \rho \mapsto \rho } from the category of smooth representations of G {\displaystyle G} to the category of non-degenerate H ( G ) {\displaystyle {\mathfrak {H}}(G)} -modules. Here, "non-degenerate" means ρ ( H ( G ) ) V = V {\displaystyle \rho ({\mathfrak {H}}(G))V=V} . The... | Wikipedia - Locally profinite group | null | null | null |
Let G ⊂ C be open with σ(T) ⊂ G. Suppose a sequence {fk} of holomorphic functions on G converges uniformly on compact subsets of G (this is sometimes called compact convergence). Then {fk(T)} is convergent in L(X): Assume for simplicity that Γ consists of only one Jordan curve. We estimate ‖ f k ( T ) − f l ( T ) ‖ = 1... | Wikipedia - Polynomial functional calculus | null | null | null |
Let Gr(r, Rn) denote the Grassmannian of r-dimensional subspaces of Rn. Let M(n, R) denote the space of real n × n matrices. Consider the set of matrices A(r, n) ⊂ M(n, R) defined by X ∈ A(r, n) if and only if the three conditions are satisfied: X is a projection operator: X2 = X. X is symmetric: Xt = X. X has trace r:... | Wikipedia - Grassmannian variety | null | null | null |
Let H ( V ) := { f: V → R } {\displaystyle {\mathcal {H}}(V):=\{f:V\rightarrow \mathbb {R} \}} be the space of (real) vertex functions. Since V {\displaystyle V} is a finite set, any vertex function f ∈ H ( V ) {\displaystyle f\in {\mathcal {H}}(V)} can be represented as a n {\displaystyle n} -dimensional vector f ∈ R ... | Wikipedia - Calculus on finite weighted graphs | null | null | null |
Furthermore, for any vertex function f ∈ H ( V ) {\displaystyle f\in {\mathcal {H}}(V)} the ℓ p {\displaystyle \ell _{p}} -norm and ℓ ∞ {\displaystyle \ell _{\infty }} -norm of f {\displaystyle f} are defined as: ‖ f ‖ p = { ( ∑ x i ∈ V | f ( x i ) | p ) 1 p , for 1 ⩽ p < ∞ , max x i ∈ V | f ( x i ) | , for p = ∞ . {\d... | Wikipedia - Calculus on finite weighted graphs | null | null | null |
Let H N = H K ⊗ H M {\displaystyle {\mathcal {H}}_{N}={\mathcal {H}}_{K}\otimes {\mathcal {H}}_{M}} be a tensor product Hilbert space. We define the product numerical radius r ⊗ ( X ) {\displaystyle r^{\otimes }(X)} of X {\displaystyle X} , with respect to this tensor product structure, as r ⊗ ( X ) = max { | z |: z ∈ ... | Wikipedia - Product numerical range | null | null | null |
Let H S {\displaystyle {\mathcal {H}}_{S}} be a finite-dimensional Hilbert space, and consider a generic (possibly mixed) quantum state ρ {\displaystyle \rho } defined on H S {\displaystyle {\mathcal {H}}_{S}} , and admitting a decomposition of the form for a collection of (not necessarily mutually orthogonal) states |... | Wikipedia - HJW theorem | null | null | null |
Furthermore, the states | Ψ S A ⟩ {\displaystyle |\Psi _{SA}\rangle } satisfying this are all and only those of the form for some orthonormal basis { | a i ⟩ } i ⊂ H A {\displaystyle \{|a_{i}\rangle \}_{i}\subset {\mathcal {H}}_{A}} . The state | Ψ S A ⟩ {\displaystyle |\Psi _{SA}\rangle } is then referred to as the "p... | Wikipedia - HJW theorem | null | null | null |
Let H be a Heyting algebra, and let F ⊆ H. We call F a filter on H if it satisfies the following properties: 1 ∈ F , {\displaystyle 1\in F,} If x , y ∈ F then x ∧ y ∈ F , {\displaystyle {\mbox{If }}x,y\in F{\mbox{ then }}x\land y\in F,} If x ∈ F , y ∈ H , and x ≤ y then y ∈ F . {\displaystyle {\mbox{If }}x\in F,\ y\in ... | Wikipedia - Heyting algebra | null | null | null |
Therefore, given any subset S of H there is a smallest filter containing S. We call it the filter generated by S. If S is empty, F = {1}. Otherwise, F is equal to the set of x in H such that there exist y1, y2, ..., yn ∈ S with y1 ∧ y2 ∧ ... ∧ yn ≤ x. If H is a Heyting algebra and F is a filter on H, we define a relati... | Wikipedia - Heyting algebra | null | null | null |
We call the Heyting algebra H/F the quotient of H by F. Let S be a subset of a Heyting algebra H and let F be the filter generated by S. Then H/F satisfies the following universal property: Given any morphism of Heyting algebras f: H → H′ satisfying f(y) = 1 for every y ∈ S, f factors uniquely through the canonical sur... | Wikipedia - Heyting algebra | null | null | null |
It is a filter on H1. (Care should be taken because this definition, if applied to a morphism of Boolean algebras, is dual to what would be called the kernel of the morphism viewed as a morphism of rings.) By the foregoing, f induces a morphism f′: H1/(ker f) → H2. It is an isomorphism of H1/(ker f) onto the subalgebra... | Wikipedia - Heyting algebra | null | null | null |
Let H be a Hopf algebra. A (left) Hopf H-module algebra A is an algebra which is a (left) module over the algebra H such that h ⋅ 1 A = ϵ ( h ) 1 A {\displaystyle h\cdot 1_{A}=\epsilon (h)1_{A}} and h ⋅ ( a b ) = ( h ( 1 ) ⋅ a ) ( h ( 2 ) ⋅ b ) {\displaystyle h\cdot (ab)=(h_{(1)}\cdot a)(h_{(2)}\cdot b)} whenever a , b... | Wikipedia - Group Hopf algebra | null | null | null |
The smash product algebra A # H {\displaystyle A\mathop {\#} H} is the vector space A ⊗ H {\displaystyle A\otimes H} with the product ( a ⊗ h ) ( b ⊗ k ) := a ( h ( 1 ) ⋅ b ) ⊗ h ( 2 ) k {\displaystyle (a\otimes h)(b\otimes k):=a(h_{(1)}\cdot b)\otimes h_{(2)}k} ,and we write a # h {\displaystyle a\mathop {\#} h} f... | Wikipedia - Group Hopf algebra | null | null | null |
Let H be a bipartite graph with parts X and Y. Let V be the set of edges of H. Let G = L(H) = the line graph of H. Then, the independence complex I ( L ( H ) ) {\displaystyle {\mathcal {I}}(L(H))} is equal to the matching complex of H, denoted M ( H ) {\displaystyle {\mathcal {M}}(H)} . It is a simplicial complex on th... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Therefore, when Meshulam's game is played, NON needs |NH(Y0)| explosions to destroy all of L(NH(Y0)), so Ψ(L(NH(Y0)) = |NH(Y0)|. Thus, Meshulam's condition Ψ ( G ) ≥ | Y 0 | {\displaystyle \Psi (G)\geq |Y_{0}|} is equivalent to Hall's marriage condition. Here, the sets Vy are pairwise-disjoint, so a C-transversal cont... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Let H be a bipartite hypergraph, and suppose C is its matching complex M ( H ) {\displaystyle {\mathcal {M}}(H)} . Let Hy (for y in Y) be sets of edges of H. For every subset Y0 of Y, M ( H Y 0 ) {\displaystyle {\mathcal {M}}(H_{Y_{0}})} is the set of matchings in the sub-hypergraph: ⋃ y ∈ Y 0 H y . {\displaystyle \big... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
This is true, in particular, if we define Hy as the set of edges of H containing the vertex y of Y. In this case, M ( H Y 0 ) {\displaystyle {\mathcal {M}}(H_{Y_{0}})} is equivalent to NH(Y0) - the multi-hypergraph of neighbors of Y0 ("multi" - since each neighbor is allowed to appear several times for several differen... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Let H = (X + Y, E) be a bipartite hypergraph. Suppose that, for every subset Y0 of Y, the following condition holds: Ψ ( L ( N H ( Y 0 ) ) ) ≥ | Y 0 | , {\displaystyle \Psi (L(N_{H}(Y_{0})))\geq |Y_{0}|,} where NH(Y0) is considered a multi-hypergraph (i.e., it may contain the same hyperedge several times, if it is a ne... | Wikipedia - Hall-type theorems for hypergraphs | null | null | null |
Let H be a semisimple finite weak Hopf algebra, then modules over H form a semisimple rigid monoidal category with finitely many simple objects. Moreover the homomorphisms spaces are finite-dimensional vector spaces and the endomorphisms space of simple objects are one-dimensional. Finally, the monoidal unit is a simpl... | Wikipedia - Weak Hopf algebra | null | null | null |
Let H be a separable infinite-dimensional Hilbert space. L(H) has two norm-closed *-ideals: I0 = {0} and the ideal K = K(H) of compact operators. Thus as a set, Prim(L(H)) = {I0, K}. Now {K} is a closed subset of Prim(L(H)). | Wikipedia - Jacobson topology | null | null | null |
The closure of {I0} is Prim(L(H)).Thus Prim(L(H)) is a non-Hausdorff space. The spectrum of L(H) on the other hand is much larger. There are many inequivalent irreducible representations with kernel K(H) or with kernel {0}. | Wikipedia - Jacobson topology | null | null | null |
Let H be a separable infinite-dimensional Hilbert space. The algebra K(H) of compact operators on H is a norm closed subalgebra of B(H). It is also closed under involution; hence it is a C*-algebra. Concrete C*-algebras of compact operators admit a characterization similar to Wedderburn's theorem for finite dimensional... | Wikipedia - C-star algebra | null | null | null |
If A is a C*-subalgebra of K(H), then there exists Hilbert spaces {Hi}i∈I such that A ≅ ⨁ i ∈ I K ( H i ) , {\displaystyle A\cong \bigoplus _{i\in I}K(H_{i}),} where the (C*-)direct sum consists of elements (Ti) of the Cartesian product Π K(Hi) with ||Ti|| → 0. Though K(H) does not have an identity element, a sequentia... | Wikipedia - C-star algebra | null | null | null |
For each natural number n let Hn be the subspace of sequences of l2 which vanish for indices k ≥ n and let en be the orthogonal projection onto Hn. The sequence {en}n is an approximate identity for K(H). K(H) is a two-sided closed ideal of B(H). For separable Hilbert spaces, it is the unique ideal. The quotient of B(H)... | Wikipedia - C-star algebra | null | null | null |
Let H be the Hilbert space completion of A {\displaystyle {\mathfrak {A}}} with respect to the inner product and let J denote the extension of the involution to a conjugate-linear involution of H. Define a representation λ and an anti-representation ρ of A {\displaystyle {\mathfrak {A}}} on itself by left and right mul... | Wikipedia - Commutation theorem for traces | null | null | null |
The proof relies on the notion of "bounded elements" in the Hilbert space completion H. An element of x in H is said to be bounded (relative to A {\displaystyle {\mathfrak {A}}} ) if the map a → xa of A {\displaystyle {\mathfrak {A}}} into H extends to a bounded operator on H, denoted by λ(x). In this case it is straig... | Wikipedia - Commutation theorem for traces | null | null | null |
Let H {\displaystyle {\mathfrak {H}}\,} be a complex, separable Hilbert space, X {\displaystyle X} a locally compact space and d ν {\displaystyle d\nu } a measure on X {\displaystyle X} . For each x {\displaystyle x} in X {\displaystyle X} , denote | x ⟩ {\displaystyle |x\rangle } a vector in H {\displaystyle {\mathfra... | Wikipedia - Coherent states in mathematical physics | null | null | null |
In order to recover the previous definition (given in the article Coherent state) of canonical or standard coherent states (CCS), it suffices to take X ≡ C {\displaystyle X\equiv \mathbb {C} } , the complex plane and d ν ( x ) ≡ 1 π d 2 x . {\textstyle d\nu (x)\equiv {\frac {1}{\pi }}d^{2}x.} | Wikipedia - Coherent states in mathematical physics | null | null | null |
Sometimes the resolution of the identity condition is replaced by a weaker condition, with the vectors | x ⟩ {\displaystyle |x\rangle } simply forming a total set in H {\displaystyle {\mathfrak {H}}\,} and the functions Ψ ( x ) = ⟨ x | ψ ⟩ {\displaystyle \Psi (x)=\langle x|\psi \rangle } , as | ψ ⟩ {\displaystyle |\psi... | Wikipedia - Coherent states in mathematical physics | null | null | null |
Let H ∗ ( X ) = H ∗ ( X , Z ) / t o r s i o n {\displaystyle H^{*}(X)=H^{*}(X,\mathbf {Z} )/\mathrm {torsion} } be the cohomology of X modulo torsion. Define the small quantum cohomology with coefficients in Λ to be Q H ∗ ( X , Λ ) = H ∗ ( X ) ⊗ Z Λ . {\displaystyle QH^{*}(X,\Lambda )=H^{*}(X)\otimes _{\mathbf {Z} }\La... | Wikipedia - Quantum cup product | null | null | null |
{\displaystyle \sum _{i}a_{i}\otimes \lambda _{i}.} The small quantum cohomology is a graded R-module with deg ( a i ⊗ λ i ) = deg ( a i ) + deg ( λ i ) . {\displaystyle \deg(a_{i}\otimes \lambda _{i})=\deg(a_{i})+\deg(\lambda _{i}).} | Wikipedia - Quantum cup product | null | null | null |
The ordinary cohomology H*(X) embeds into QH*(X, Λ) via a ↦ a ⊗ 1 {\displaystyle a\mapsto a\otimes 1} , and QH*(X, Λ) is generated as a Λ-module by H*(X). For any two cohomology classes a, b in H*(X) of pure degree, and for any A in H 2 ( X ) {\displaystyle H_{2}(X)} , define (a∗b)A to be the unique element of H*(X) su... | Wikipedia - Quantum cup product | null | null | null |
(The right-hand side is a genus-0, 3-point Gromov–Witten invariant.) Then define a ∗ b := ∑ A ∈ H 2 ( X ) ( a ∗ b ) A ⊗ e A . {\displaystyle a*b:=\sum _{A\in H_{2}(X)}(a*b)_{A}\otimes e^{A}.} This extends by linearity to a well-defined Λ-bilinear map Q H ∗ ( X , Λ ) ⊗ Q H ∗ ( X , Λ ) → Q H ∗ ( X , Λ ) {\displaystyle QH... | Wikipedia - Quantum cup product | null | null | null |
Let Hn(R) be the space of real symmetric n by n matrices with inner product (a,b) = Tr ab and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(R) is a simple Euclidean Jordan algebra of rank n for n ≥ 3. Let Hn(C) be the space of complex self-adjoint n by n matrices with inner product (a,b) = Re Tr ab* and Jordan product a... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let Hn(H) be the space of self-adjoint n by n matrices with entries in the quaternions, inner product (a,b) = Re Tr ab* and Jordan product a ∘ b = 1/2(ab + ba). Then Hn(H) is a simple Euclidean Jordan algebra of rank n for n ≥ 3. | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let V be a finite dimensional real inner product space and set E = V ⊕ R with inner product (u⊕λ,v⊕μ) =(u,v) + λμ and product (u⊕λ)∘(v⊕μ)=( μu + λv) ⊕ . This is a Euclidean Jordan algebra of rank 2, called a spin factor. The above examples in fact give all the simple Euclidean Jordan algebras, except for one exceptiona... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let I ( X ) {\displaystyle \mathrm {I} (X)} be the ideal of Laurent polynomials that vanish on X in K {\displaystyle K} . Define Trop ( X ) = ⋂ f ∈ I ( X ) V ( Trop ( f ) ) ⊆ R n . {\displaystyle \operatorname {Trop} (X)=\bigcap _{f\in \mathrm {I} (X)}\mathrm {V} (\operatorname {Trop} (f))\subseteq \mathbb {R} ^{n... | Wikipedia - Tropical geometry | null | null | null |
Every tropical variety is the intersection of a finite number of tropical hypersurfaces. A finite set of polynomials { f 1 , … , f r } ⊆ I ( X ) {\displaystyle \{f_{1},\ldots ,f_{r}\}\subseteq \mathrm {I} (X)} is called a tropical basis for X if Trop ( X ) {\displaystyle \operatorname {Trop} (X)} is the intersection ... | Wikipedia - Tropical geometry | null | null | null |
Let I = { a , b , c , … } {\displaystyle I=\{{\mathsf {a,b,c,}}\ldots \}} be an "alphabet", namely a (usually finite) set of objects called "letters". Let W denote the corresponding set of words or "strings", which will be denoted as in strings, namely either by writing their letters in sequence or by ϵ {\displaystyle ... | Wikipedia - Basis (universal algebra) | null | null | null |
(Depending on the set-theoretical implementation of sequences, b may not be an identity function, namely i {\displaystyle i} may not be i {\displaystyle {\mathsf {i}}} , rather an object like { ( ∅ , i ) } {\displaystyle \{(\emptyset ,{\mathsf {i}})\}} , namely a singleton function, or a pair like ( ∅ , i ) {\displayst... | Wikipedia - Basis (universal algebra) | null | null | null |
Let I be an ideal of A and define the "double" to be a subring of the Cartesian product A×A: D ( A , I ) = { ( x , y ) ∈ A × A: x − y ∈ I } . {\displaystyle D(A,I)=\{(x,y)\in A\times A:x-y\in I\}\ .} The relative K-group is defined in terms of the "double" K 0 ( A , I ) = ker ( K 0 ( D ( A , I ) ) → K 0 ( A ) ) . | Wikipedia - Special Whitehead group | null | null | null |
{\displaystyle K_{0}(A,I)=\ker \left({K_{0}(D(A,I))\rightarrow K_{0}(A)}\right)\ .} where the map is induced by projection along the first factor. The relative K0(A,I) is isomorphic to K0(I), regarding I as a ring without identity. The independence from A is an analogue of the Excision theorem in homology. | Wikipedia - Special Whitehead group | null | null | null |
Let I be the set of interpretations of a Datalog program P, that is, I = P(H), where H is the Herbrand base of P and P is the powerset operator. The immediate consequence operator for P is the following map T from I to I: For each ground instance of each rule in P, if every clause in the body is in the input interpreta... | Wikipedia - Syntax and semantics of logic programming | null | null | null |
Let I denote the identity matrix and let J denote the matrix of ones, both matrices of order v. The adjacency matrix A of a strongly regular graph satisfies two equations. First: A J = J A = k J , {\displaystyle AJ=JA=kJ,} which is a restatement of the regularity requirement. This shows that k is an eigenvalue of the a... | Wikipedia - Strongly regular graph | null | null | null |
The ij-th element of the left hand side gives the number of two-step paths from i to j. The first term of the right hand side gives the number of two-step paths from i back to i, namely k edges out and back in. The second term gives the number of two-step paths when i and j are directly connected. The third term gives ... | Wikipedia - Strongly regular graph | null | null | null |
Let I t {\displaystyle I_{t}} represent the number of cases of infection at time t {\displaystyle t} . Assume all cases recover or are removed in exactly one time-step. Let S t {\displaystyle S_{t}} represent the number of susceptible individuals at time t {\displaystyle t} . | Wikipedia - Reed–Frost model | null | null | null |
Let B ( x ) {\displaystyle {\mathcal {B}}(x)} be a Bernoulli random variable that returns 1 {\displaystyle 1} with probability x {\displaystyle x} and 0 {\displaystyle 0} with probability 1 − x {\displaystyle 1-x} . Making use of the random-variable multiplication convention, we can write the Reed–Frost model as I t + ... | Wikipedia - Reed–Frost model | null | null | null |
Let I ⊂ R be an open interval and let γ: I → R2 be a smooth plane curve parametrised by arc length. Consider the one-parameter family of normal lines to γ(I). A line is normal to γ at γ(t) if it passes through γ(t) and is perpendicular to the tangent vector to γ at γ(t). Let T denote the unit tangent vector to γ and le... | Wikipedia - Envelope (mathematics) | null | null | null |
Using a dot to denote the dot product, the generating family for the one-parameter family of normal lines is given by F: I × R2 → R where F ( t , x ) = ( x − γ ( t ) ) ⋅ T ( t ) . {\displaystyle F(t,{\mathbf {x} })=({\mathbf {x} }-\gamma (t))\cdot {\mathbf {T} }(t)\ .} Clearly (x − γ)·T = 0 if and only if x − γ is perp... | Wikipedia - Envelope (mathematics) | null | null | null |
To find the discriminant of F we need to compute its partial derivative with respect to t: ∂ F ∂ t ( t , x ) = κ ( t ) ( x − γ ( t ) ) ⋅ N ( t ) − 1 , {\displaystyle {\frac {\partial F}{\partial t}}(t,{\mathbf {x} })=\kappa (t)({\mathbf {x} }-\gamma (t))\cdot {\mathbf {N} }(t)-1\ ,} where κ is the plane curve curvature... | Wikipedia - Envelope (mathematics) | null | null | null |
Assuming that κ ≠ 0 it follows that λ = 1/κ and so D = γ ( t ) + 1 κ ( t ) N ( t ) . {\displaystyle {\mathcal {D}}=\gamma (t)+{\frac {1}{\kappa (t)}}{\mathbf {N} }(t)\ .} This is exactly the evolute of the curve γ. | Wikipedia - Envelope (mathematics) | null | null | null |
Let K = ( T , A ) {\displaystyle {\mathcal {K}}=({\mathcal {T}},{\mathcal {A}})} be a knowledge base. I ⊨ K {\displaystyle {\mathcal {I}}\models {\mathcal {K}}} if and only if I ⊨ T {\displaystyle {\mathcal {I}}\models {\mathcal {T}}} and I ⊨ A {\displaystyle {\mathcal {I}}\models {\mathcal {A}}} | Wikipedia - Description Logics | null | null | null |
Let K = F(x), the rational functions on the affine line X = F1, and take a point a ∈ X. For a polynomial f ( x ) = a k ( x − a ) k + a k + 1 ( x − a ) k + 1 + ⋯ + a n ( x − a ) n {\displaystyle f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}} with a k ≠ 0 {\displaystyle a_{k}\neq 0} , define va(f) =... | Wikipedia - Equivalence of valuations | null | null | null |
Let K be a closed convex subset of a real vector space V and ∂K be the boundary of K. The solid tangent cone to K at a point x ∈ ∂K is the closure of the cone formed by all half-lines (or rays) emanating from x and intersecting K in at least one point y distinct from x. It is a convex cone in V and can also be defined ... | Wikipedia - Clarke tangent cone | null | null | null |
Let K be a commutative ring. In most applications, K is a field of characteristic 0, such as R or C. A superalgebra over K is a K-module A with a direct sum decomposition A = A 0 ⊕ A 1 {\displaystyle A=A_{0}\oplus A_{1}} together with a bilinear multiplication A × A → A such that A i A j ⊆ A i + j {\displaystyle A_{i}A... | Wikipedia - Superalgebra | null | null | null |
Elements of parity 0 are said to be even and those of parity 1 to be odd. If x and y are both homogeneous then so is the product xy and | x y | = | x | + | y | {\displaystyle |xy|=|x|+|y|} . An associative superalgebra is one whose multiplication is associative and a unital superalgebra is one with a multiplicative ide... | Wikipedia - Superalgebra | null | null | null |
The identity element in a unital superalgebra is necessarily even. Unless otherwise specified, all superalgebras in this article are assumed to be associative and unital. A commutative superalgebra (or supercommutative algebra) is one which satisfies a graded version of commutativity. Specifically, A is commutative if ... | Wikipedia - Superalgebra | null | null | null |
Let K be a field lying between Q and its p-adic completion Qp with respect to the usual non-Archimedean p-adic norm ‖x‖p on Q for some prime p. Let R be the subring of K defined by R = { x: ‖x‖p ≤ 1 } When K = Q, R is the localization of Z at p and, when K = Qp, R = Zp, the p-adic integers, i.e. the closure of Z in Qp.... | Wikipedia - Building (mathematics) | null | null | null |
By definition each apartment has the required form and their union is the whole of X. The second axiom follows by a variant of the Schreier refinement argument. The last axiom follows by a simple counting argument based on the orders of finite Abelian groups of the form L + pk ·Li / pk ·LiA standard compactness argumen... | Wikipedia - Building (mathematics) | null | null | null |
The building comes equipped with a labelling of its vertices with values in Z / nZ. Indeed, fixing a reference lattice L, the label of M is given by label(M) = logp |M / pk L| modulo nfor k sufficiently large. The vertices of any (n – 1)-simplex in X has distinct labels, running through the whole of Z / nZ. Any simplic... | Wikipedia - Building (mathematics) | null | null | null |
Let K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic set V is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring R = K . {\displaystyle R=K.} | Wikipedia - Dimension of an algebraic variety | null | null | null |
Let A = R / I {\displaystyle A=R/I} be the algebra of the polynomial functions over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros (that is having the... | Wikipedia - Dimension of an algebraic variety | null | null | null |
The dimension of V is The maximal length d {\displaystyle d} of the chains V 0 ⊂ V 1 ⊂ … ⊂ V d {\displaystyle V_{0}\subset V_{1}\subset \ldots \subset V_{d}} of distinct nonempty (irreducible) subvarieties of V.This definition generalizes a property of the dimension of a Euclidean space or a vector space. It is thus pr... | Wikipedia - Dimension of an algebraic variety | null | null | null |
If V {\displaystyle V} is irreducible, it turns out that all the local rings at closed points have the same Krull dimension (see ). If V is a variety, the Krull dimension of the local ring at any point of VThis rephrases the previous definition into a more geometric language. The maximal dimension of the tangent vector... | Wikipedia - Dimension of an algebraic variety | null | null | null |
More precisely, if V if defined over the reals, then the set of its real regular points, if it is not empty, is a differentiable manifold that has the same dimension as a variety and as a manifold. If V is a variety, the dimension of the tangent vector space at any non singular point of V.This is the algebraic analogue... | Wikipedia - Dimension of an algebraic variety | null | null | null |
The number of hyperplanes or hypersurfaces in general position which are needed to have an intersection with V which is reduced to a nonzero finite number of points.This definition is not intrinsic as it apply only to algebraic sets that are explicitly embedded in an affine or projective space. The maximal length of a ... | Wikipedia - Dimension of an algebraic variety | null | null | null |
Moreover, the dimension is not changed if the polynomials of the Gröbner basis are replaced with their leading monomials, and if these leading monomials are replaced with their radical (monomials obtained by removing exponents). So: The Krull dimension of the Stanley–Reisner ring R / J {\displaystyle R/J} where J {\dis... | Wikipedia - Dimension of an algebraic variety | null | null | null |
Let K be a field, and let A be a vector space over K equipped with an additional binary operation from A × A to A, denoted here by · (that is, if x and y are any two elements of A, then x · y is an element of A that is called the product of x and y). Then A is an algebra over K if the following identities hold for all ... | Wikipedia - Algebra over a field | null | null | null |
Let K be a field. Let L be a commutative unital associative K-algebra. Then L is called an étale K-algebra if any one of the following equivalent conditions holds: | Wikipedia - Étale algebra | null | null | null |
Let K be a local field with valuation v and D a K-algebra. We may assume D is a division algebra with centre K of degree n. The valuation v can be extended to D, for example by extending it compatibly to each commutative subfield of D: the value group of this valuation is (1/n)Z.There is a commutative subfield L of D w... | Wikipedia - Hasse invariant of an algebra | null | null | null |
The invariant map attaches the element k/n mod 1 to the class. This exhibits the invariant map as a homomorphism inv L / K: Br ( L / K ) → Q / Z . {\displaystyle {\underset {L/K}{\operatorname {inv} }}:\operatorname {Br} (L/K)\rightarrow \mathbb {Q} /\mathbb {Z} .} The invariant map extends to Br(K) by representing e... | Wikipedia - Hasse invariant of an algebra | null | null | null |
Let K {\displaystyle K} be a (commutative) field and A = K {\displaystyle A=K} be a commutative polynomial ring (with A = K {\displaystyle A=K} when s = 0 {\displaystyle s=0} ). The iterated skew polynomial ring A ⋯ {\displaystyle A\cdots } is called an Ore algebra when the σ i {\displaystyle \sigma _{i}} and δ j {\... | Wikipedia - Ore algebra | null | null | null |
Let K {\displaystyle K} be a field and V {\displaystyle V} a vector space over K {\displaystyle K} . A mapping q {\displaystyle q} from V {\displaystyle V} to K {\displaystyle K} such that (Q1) q ( λ x → ) = λ 2 q ( x → ) {\displaystyle \;q(\lambda {\vec {x}})=\lambda ^{2}q({\vec {x}})\;} for any λ ∈ K {\displaystyle \... | Wikipedia - Quadric | null | null | null |
Let K {\displaystyle K} be a field, for instance the complex numbers, and A {\displaystyle {\mathcal {A}}} be a K {\displaystyle K} -algebra (i.e. a vector space over K {\displaystyle K} with a binary operation ∗: A ⊗ A → A {\displaystyle \ast :{\mathcal {A}}\otimes {\mathcal {A}}\to {\mathcal {A}}} that is linear in b... | Wikipedia - Algebraic signal processing | null | null | null |
To be an algebra homomorphism, ρ {\displaystyle \rho } must not only be a linear transformation, but also satisfy the propertyGiven a signal x ∈ M {\displaystyle x\in {\mathcal {M}}} , convolution of the signal by a filter a ∈ A {\displaystyle a\in {\mathcal {A}}} yields a new signal ρ a ( x ) {\displaystyle \rho _{a}(... | Wikipedia - Algebraic signal processing | null | null | null |
The image of a generator g ∈ G {\displaystyle g\in {\mathcal {G}}} is called a shift operator. In all practically all examples, convolutions are formed as polynomials in E n d ( M ) {\displaystyle \mathrm {End} ({\mathcal {M}})} generated by shift operators. However, this is not necessarily the case for a representatio... | Wikipedia - Algebraic signal processing | null | null | null |
Let K {\displaystyle K} be a near field. Let K m {\displaystyle K_{m}} be its multiplicative group and let K a {\displaystyle K_{a}} be its additive group. Let c ∈ K m {\displaystyle c\in K_{m}} act on b ∈ K a {\displaystyle b\in K_{a}} by b ↦ b ⋅ c {\displaystyle b\mapsto b\cdot c} . The axioms of a near field show th... | Wikipedia - Near-field (mathematics) | null | null | null |
Conversely, if A {\displaystyle A} is an abelian group and M {\displaystyle M} is a subgroup of A u t ( A ) {\displaystyle \mathrm {Aut} (A)} which acts freely and transitively on the nonzero elements of A {\displaystyle A} , then we can define a near field with additive group A {\displaystyle A} and multiplicative gro... | Wikipedia - Near-field (mathematics) | null | null | null |
Then we define addition on A {\displaystyle A} by the additive group structure on A {\displaystyle A} and define multiplication by a ⋅ b = 1 ∗ ϕ − 1 ( a ) ϕ − 1 ( b ) {\displaystyle a\cdot b=1\ast \phi ^{-1}(a)\phi ^{-1}(b)} . A Frobenius group can be defined as a finite group of the form A ⋊ M {\displaystyle A\rtimes ... | Wikipedia - Near-field (mathematics) | null | null | null |
Let K {\displaystyle K} be a space of real-valued functions, with the l-infinity metric (see example 3 above). Suppose all functions in K {\displaystyle K} are bounded by a real constant M {\displaystyle M} . Then, the covering number can be used to bound the generalization error of learning functions from K {\displays... | Wikipedia - Covering number | null | null | null |
In this case, the ideal a {\displaystyle {\mathfrak {a}}} and its class a P K {\displaystyle {\mathfrak {a}}{\mathcal {P}}_{K}} are said to principalize or capitulate in L {\displaystyle L} . This phenomenon is described most conveniently by the principalization kernel or capitulation kernel, that is the kernel ker (... | Wikipedia - Principalization (algebra) | null | null | null |
Let K {\textstyle \mathbb {K} } be a field of characteristic zero. A nonzero sequence y ( n ) {\textstyle y(n)} is called hypergeometric if the ratio of two consecutive terms is rational, i.e. y ( n + 1 ) / y ( n ) ∈ K ( n ) {\textstyle y(n+1)/y(n)\in \mathbb {K} (n)} . The Petkovšek algorithm uses as key concept that ... | Wikipedia - Petkovšek's algorithm | null | null | null |
Then there exist monic polynomials a , b , c ∈ K {\textstyle a,b,c\in \mathbb {K} } and 0 ≠ z ∈ K {\textstyle 0\neq z\in \mathbb {K} } such that r ( n ) = z a ( n ) b ( n ) c ( n + 1 ) c ( n ) {\displaystyle r(n)=z{\frac {a(n)}{b(n)}}{\frac {c(n+1)}{c(n)}}} and gcd ( a ( n ) , b ( n + k ) ) = 1 {\textstyle \gcd(a(n),b... | Wikipedia - Petkovšek's algorithm | null | null | null |
Let K ⊂ P 3 {\displaystyle K\subset \mathbb {P} ^{3}} be a quartic surface with an ordinary double point p, near which K looks like a quadratic cone. Any projective line through p then meets K with multiplicity two at p, and will therefore meet the quartic K in just two other points. Identifying the lines in P 3 {\disp... | Wikipedia - Kummer surface | null | null | null |
Hence the maximal number of nodes on a quartic is 16, and in this case they are all simple nodes (to show that p {\displaystyle p} is simple project from another node). A quartic which obtains these 16 nodes is called a Kummer Quartic, and we will concentrate on them below. Since p {\displaystyle p} is a simple node, t... | Wikipedia - Kummer surface | null | null | null |
This conic is in fact tangent to the six lines (w.o proof). Conversely, given a configuration of a conic and six lines which tangent to it in the plane, we may define the double cover of the plane ramified over the union of these 6 lines. This double cover may be mapped to P 3 {\displaystyle \mathbb {P} ^{3}} , under a... | Wikipedia - Kummer surface | null | null | null |
Let L / K {\displaystyle L/K} be a field extension. The set L {\displaystyle L} becomes a pregeometry if we define cl ( A ) = { x ∈ L: x is algebraic over K ( A ) } {\displaystyle {\text{cl}}(A)=\{x\in L:x{\text{ is algebraic over }}K(A)\}} for A ⊆ L {\displaystyle A\subseteq L} . The set A {\displaystyle A} is indepen... | Wikipedia - Pregeometry (model theory) | null | null | null |
The dimension of A {\displaystyle A} coincides with the transcendence degree trdeg ( K ( A ) / K ) {\displaystyle {\text{trdeg}}(K(A)/K)} . In model theory, the case of L {\displaystyle L} being algebraically closed and K {\displaystyle K} its prime field is especially important. While vector spaces are modular and aff... | Wikipedia - Pregeometry (model theory) | null | null | null |
Let L 1 = L ( A , Ω , Z , I ) {\displaystyle {\mathcal {L}}_{1}={\mathcal {L}}(\mathrm {A} ,\Omega ,\mathrm {Z} ,\mathrm {I} )} , where A {\displaystyle \mathrm {A} } , Ω {\displaystyle \Omega } , Z {\displaystyle \mathrm {Z} } , I {\displaystyle \mathrm {I} } are defined as follows: The set A {\displaystyle \mathrm {A... | Wikipedia - Truth-functional propositional calculus | null | null | null |
Alternatively, all of the logical operators may be defined in terms of a sole sufficient operator, such as the Sheffer stroke (nand). The biconditional ( a ↔ b {\displaystyle a\leftrightarrow b} ) can of course be defined in terms of conjunction and implication as ( a → b ) ∧ ( b → a ) {\displaystyle (a\to b)\land (b\t... | Wikipedia - Truth-functional propositional calculus | null | null | null |
{\displaystyle \Omega _{2}=\{\to \}.} Then a ∨ b {\displaystyle a\lor b} is defined as ¬ a → b {\displaystyle \neg a\to b} , and a ∧ b {\displaystyle a\land b} is defined as ¬ ( a → ¬ b ) {\displaystyle \neg (a\to \neg b)} . The set I {\displaystyle \mathrm {I} } (the set of initial points of logical deduction, i.e., l... | Wikipedia - Truth-functional propositional calculus | null | null | null |
Let L 2 = L ( A , Ω , Z , I ) {\displaystyle {\mathcal {L}}_{2}={\mathcal {L}}(\mathrm {A} ,\Omega ,\mathrm {Z} ,\mathrm {I} )} , where A {\displaystyle \mathrm {A} } , Ω {\displaystyle \Omega } , Z {\displaystyle \mathrm {Z} } , I {\displaystyle \mathrm {I} } are defined as follows: The alpha set A {\displaystyle \mat... | Wikipedia - Sentential calculus | null | null | null |
{\displaystyle \Omega _{2}=\{\land ,\lor ,\to ,\leftrightarrow \}.} In the following example of a propositional calculus, the transformation rules are intended to be interpreted as the inference rules of a so-called natural deduction system. The particular system presented here has no initial points, which means that i... | Wikipedia - Sentential calculus | null | null | null |
The set of initial points is empty, that is, I = ∅ {\displaystyle \mathrm {I} =\varnothing } . The set of transformation rules, Z {\displaystyle \mathrm {Z} } , is described as follows:Our propositional calculus has eleven inference rules. These rules allow us to derive other true formulas given a set of formulas that ... | Wikipedia - Sentential calculus | null | null | null |
The first ten simply state that we can infer certain well-formed formulas from other well-formed formulas. The last rule however uses hypothetical reasoning in the sense that in the premise of the rule we temporarily assume an (unproven) hypothesis to be part of the set of inferred formulas to see if we can infer a cer... | Wikipedia - Sentential calculus | null | null | null |
In describing the transformation rules, we may introduce a metalanguage symbol ⊢ {\displaystyle \vdash } . It is basically a convenient shorthand for saying "infer that". | Wikipedia - Sentential calculus | null | null | null |
The format is Γ ⊢ ψ {\displaystyle \Gamma \vdash \psi } , in which Γ is a (possibly empty) set of formulas called premises, and ψ is a formula called conclusion. The transformation rule Γ ⊢ ψ {\displaystyle \Gamma \vdash \psi } means that if every proposition in Γ is a theorem (or has the same truth value as the axioms... | Wikipedia - Sentential calculus | null | null | null |
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